Optimal. Leaf size=92 \[ -\frac {i A+B}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i A \sqrt {c-i c \tan (e+f x)}}{c f \sqrt {a+i a \tan (e+f x)}} \]
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Rubi [A]
time = 0.13, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3669, 79, 37}
\begin {gather*} \frac {i A \sqrt {c-i c \tan (e+f x)}}{c f \sqrt {a+i a \tan (e+f x)}}-\frac {B+i A}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {(a A) \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i A \sqrt {c-i c \tan (e+f x)}}{c f \sqrt {a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.10, size = 77, normalized size = 0.84 \begin {gather*} -\frac {(\cos (e+f x)+i \sin (e+f x)) (B \cos (e+f x)-A \sin (e+f x)) \sqrt {c-i c \tan (e+f x)}}{c f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 99, normalized size = 1.08
method | result | size |
risch | \(-\frac {i A \,{\mathrm e}^{2 i \left (f x +e \right )}+B \,{\mathrm e}^{2 i \left (f x +e \right )}-i A +B}{2 \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(92\) |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (A \left (\tan ^{3}\left (f x +e \right )\right )-B \left (\tan ^{2}\left (f x +e \right )\right )+A \tan \left (f x +e \right )-B \right )}{f a c \left (i+\tan \left (f x +e \right )\right )^{2} \left (i-\tan \left (f x +e \right )\right )^{2}}\) | \(99\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (A \left (\tan ^{3}\left (f x +e \right )\right )-B \left (\tan ^{2}\left (f x +e \right )\right )+A \tan \left (f x +e \right )-B \right )}{f a c \left (i+\tan \left (f x +e \right )\right )^{2} \left (i-\tan \left (f x +e \right )\right )^{2}}\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, size = 132, normalized size = 1.43 \begin {gather*} -\frac {2 \, {\left ({\left (A - i \, B\right )} \cos \left (4 \, f x + 4 \, e\right ) - 2 i \, B \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - B\right )} \sin \left (4 \, f x + 4 \, e\right ) + 2 \, B \sin \left (2 \, f x + 2 \, e\right ) - A - i \, B\right )} \sqrt {a} \sqrt {c}}{-4 \, {\left (i \, a c \cos \left (3 \, f x + 3 \, e\right ) + i \, a c \cos \left (f x + e\right ) - a c \sin \left (3 \, f x + 3 \, e\right ) - a c \sin \left (f x + e\right )\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.15, size = 121, normalized size = 1.32 \begin {gather*} \frac {{\left ({\left (-i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, B e^{\left (3 i \, f x + 3 i \, e\right )} - 2 \, B e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, B e^{\left (i \, f x + i \, e\right )} + i \, A - B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-i \, f x - i \, e\right )}}{2 \, a c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B \tan {\left (e + f x \right )}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 143, normalized size = 1.55 \begin {gather*} -\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,1{}\mathrm {i}+B-A\,\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+B\,\cos \left (2\,e+2\,f\,x\right )-A\,\sin \left (2\,e+2\,f\,x\right )-B\,\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{2\,a\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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